Cross Product Two Vectors

Given two linearly independent vectors a and b, the cross product, a b (read "a cross b"), is a vector that is perpendicular to both a and b, [1] and thus normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming. When two vectors are multiplied with each other and the product of the vectors is also a vector quantity, then the resultant vector is called the cross product of two vectors or the vector product.

A vector has magnitude (how long it is) and direction: Two vectors can be multiplied using the Cross Product (also see Dot Product). The cross product, also known as the vector product, is a binary operation that takes two vectors in a three-dimensional Euclidean space and produces another vector. The cross product is very useful for several types of calculations, including finding a vector orthogonal to two given vectors, computing areas of triangles and parallelograms, and even determining the volume of the three-dimensional geometric shape made of parallelograms known as a parallelepiped.

In this section we define the cross product of two vectors and give some of the basic facts and properties of cross products. Instead of thinking When do I need the cross product? think When do I need interactions between different dimensions?. Area, for example, is formed by vectors pointing in different directions (the more orthogonal, the better).

Indeed, the cross product measures the area spanned by two 3d vectors (source): Since the cross product must be perpendicular to the two unit vectors, it must be equal to the other unit vector or the opposite of that unit vector. Looking at the above graph, you can use the right-hand rule to determine the following results.

Cross product, a method of multiplying two vectors that produces a vector perpendicular to both vectors involved in the multiplication; that is, a b = c, where c is perpendicular to both a and b. Check out our article and learn how to calculate the cross product of two vectors. You will see several aspects of vector algebra, including the right-hand rule and the difference between the cross product and the dot product.